An excellent review of the influence of various tSNE parameters can be found here: https://distill.pub/2016/misread-tsne/.
The tSNE algorithm attempts to prioritise local relationships between data points over global relationships. As a result, similar cells will clump together into ‘islands’, but two separate islands that are near each other on the 2D tSNE plots are not necessarily similar to each other.
One of the values that is used to strike this balance is perplexity. Conceptually, perplexity helps decide how large the 'local’ relationships will be in n-dimensional space.
A low perplexity value will result in the algorithm looking for minute differences between small groups of cells. This helps us to understand the underlying structure of the data by enabling us to visualise complex relationships in high-dimensional data. However, a perplexity value that is too low may create islands of cells that separate from each other in tSNE space due to noise or irrelevant variance, but are in reality very similar to each other.
A very large perplexity value will result in the algorithm trying to balance the relationship of a large number of data points. This helps to prevent creating artificial divisions of very similar groups of cells. However, a perplexity value that is too high will recreate much of the overall data structure, at the cost of maintaining local relationships.
An example of the influence of the perplexity value can be found here: https://distill.pub/2016/misread-tsne/.
In this example, perplexity = 2 is too low, and artificial divisions are being generated. Perplexity values between 5 and 50 largely capture the structure of the data, although some artificial variance is evident for perplexity = 5.
Another excellent example of how the perplexity parameter will influence tSNE is provided here: https://stats.stackexchange.com/questions/263539/clustering-on-the-output-of-t-sne.
Here is another example from https://stats.stackexchange.com/questions/263539/clustering-on-the-output-of-t-sne.